oppc1stf - Mathbox for Zhi Wang

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Theorem oppc1stf 49320

Description: The opposite functor of the first projection functor is the first projection functor of opposite categories. (Contributed by Zhi Wang, 19-Nov-2025.)

**Hypotheses**
Ref	Expression
oppc1stf.o	⊢ 𝑂 = (oppCat‘𝐶)
oppc1stf.p	⊢ 𝑃 = (oppCat‘𝐷)
oppc1stf.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
oppc1stf.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

**Assertion**
Ref	Expression
oppc1stf	⊢ (𝜑 → ( oppFunc ‘(𝐶 1^st_F 𝐷)) = (𝑂 1^st_F 𝑃))

Proof of Theorem oppc1stf Dummy variables 𝑥 𝑦 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppc1stf.o	. 2 ⊢ 𝑂 = (oppCat‘𝐶)
2		oppc1stf.p	. 2 ⊢ 𝑃 = (oppCat‘𝐷)
3		oppc1stf.c	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
4		oppc1stf.d	. 2 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
5		eqid 2731	. . . . . 6 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))) = (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))
6	5	tposmpo 8188	. . . . 5 ⊢ tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))) = (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))
7		eqid 2731	. . . . . . . . . 10 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
8	7, 1	oppchom 17616	. . . . . . . . 9 ⊢ ((1^st ‘𝑦)(Hom ‘𝑂)(1^st ‘𝑥)) = ((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦))
9		eqid 2731	. . . . . . . . . 10 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10	9, 2	oppchom 17616	. . . . . . . . 9 ⊢ ((2^nd ‘𝑦)(Hom ‘𝑃)(2^nd ‘𝑥)) = ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦))
11	8, 10	xpeq12i 5639	. . . . . . . 8 ⊢ (((1^st ‘𝑦)(Hom ‘𝑂)(1^st ‘𝑥)) × ((2^nd ‘𝑦)(Hom ‘𝑃)(2^nd ‘𝑥))) = (((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦)))
12		eqid 2731	. . . . . . . . 9 ⊢ (𝑂 ×_c 𝑃) = (𝑂 ×_c 𝑃)
13		eqid 2731	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
14	1, 13	oppcbas 17619	. . . . . . . . . 10 ⊢ (Base‘𝐶) = (Base‘𝑂)
15		eqid 2731	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
16	2, 15	oppcbas 17619	. . . . . . . . . 10 ⊢ (Base‘𝐷) = (Base‘𝑃)
17	12, 14, 16	xpcbas 18079	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝑂 ×_c 𝑃))
18		eqid 2731	. . . . . . . . 9 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
19		eqid 2731	. . . . . . . . 9 ⊢ (Hom ‘𝑃) = (Hom ‘𝑃)
20		eqid 2731	. . . . . . . . 9 ⊢ (Hom ‘(𝑂 ×_c 𝑃)) = (Hom ‘(𝑂 ×_c 𝑃))
21		simp2 1137	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)))
22		simp3 1138	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)))
23	12, 17, 18, 19, 20, 21, 22	xpchom 18081	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥) = (((1^st ‘𝑦)(Hom ‘𝑂)(1^st ‘𝑥)) × ((2^nd ‘𝑦)(Hom ‘𝑃)(2^nd ‘𝑥))))
24		eqid 2731	. . . . . . . . 9 ⊢ (𝐶 ×_c 𝐷) = (𝐶 ×_c 𝐷)
25	24, 13, 15	xpcbas 18079	. . . . . . . . 9 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘(𝐶 ×_c 𝐷))
26		eqid 2731	. . . . . . . . 9 ⊢ (Hom ‘(𝐶 ×_c 𝐷)) = (Hom ‘(𝐶 ×_c 𝐷))
27	24, 25, 7, 9, 26, 22, 21	xpchom 18081	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦) = (((1^st ‘𝑥)(Hom ‘𝐶)(1^st ‘𝑦)) × ((2^nd ‘𝑥)(Hom ‘𝐷)(2^nd ‘𝑦))))
28	11, 23, 27	3eqtr4a 2792	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥) = (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))
29	28	reseq2d 5923	. . . . . 6 ⊢ (((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) ∧ 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ∧ 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷))) → (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥)) = (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))
30	29	mpoeq3dva 7418	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥))) = (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))))
31	6, 30	eqtr4id 2785	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦))) = (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥))))
32	31	opeq2d 4827	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))⟩ = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥)))⟩)
33		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝐶 ∈ Cat)
34		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝐷 ∈ Cat)
35		eqid 2731	. . . . 5 ⊢ (𝐶 1^st_F 𝐷) = (𝐶 1^st_F 𝐷)
36	24, 25, 26, 33, 34, 35	1stfval 18092	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝐶 1^st_F 𝐷) = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))⟩)
37	24, 33, 34, 35	1stfcl 18098	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝐶 1^st_F 𝐷) ∈ ((𝐶 ×_c 𝐷) Func 𝐶))
38	36, 37	oppfval3 49170	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ( oppFunc ‘(𝐶 1^st_F 𝐷)) = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), tpos (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑥(Hom ‘(𝐶 ×_c 𝐷))𝑦)))⟩)
39	1	oppccat 17623	. . . . 5 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
40	33, 39	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝑂 ∈ Cat)
41	2	oppccat 17623	. . . . 5 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
42	34, 41	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → 𝑃 ∈ Cat)
43		eqid 2731	. . . 4 ⊢ (𝑂 1^st_F 𝑃) = (𝑂 1^st_F 𝑃)
44	12, 17, 20, 40, 42, 43	1stfval 18092	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → (𝑂 1^st_F 𝑃) = ⟨(1^st ↾ ((Base‘𝐶) × (Base‘𝐷))), (𝑦 ∈ ((Base‘𝐶) × (Base‘𝐷)), 𝑥 ∈ ((Base‘𝐶) × (Base‘𝐷)) ↦ (1^st ↾ (𝑦(Hom ‘(𝑂 ×_c 𝑃))𝑥)))⟩)
45	32, 38, 44	3eqtr4d 2776	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat)) → ( oppFunc ‘(𝐶 1^st_F 𝐷)) = (𝑂 1^st_F 𝑃))
46		df-1stf 18074	. 2 ⊢ 1^st_F = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ ⦋((Base‘𝑐) × (Base‘𝑑)) / 𝑏⦌⟨(1^st ↾ 𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1^st ↾ (𝑥(Hom ‘(𝑐 ×_c 𝑑))𝑦)))⟩)
47	1, 2, 3, 4, 45, 46	oppc1stflem 49319	1 ⊢ (𝜑 → ( oppFunc ‘(𝐶 1^st_F 𝐷)) = (𝑂 1^st_F 𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ⦋csb 3845 ⟨cop 4577 × cxp 5609 ↾ cres 5613 ‘cfv 6476 (class class class)co 7341 ∈ cmpo 7343 1^st c1st 7914 2^nd c2nd 7915 tpos ctpos 8150 Basecbs 17115 Hom chom 17167 Catccat 17565 oppCatcoppc 17612 ×_c cxpc 18069 1^st_F c1stf 18070 oppFunc coppf 49154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-map 8747 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-fz 13403 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-hom 17180 df-cco 17181 df-cat 17569 df-cid 17570 df-homf 17571 df-comf 17572 df-oppc 17613 df-func 17760 df-xpc 18073 df-1stf 18074 df-oppf 49155

This theorem is referenced by: (None)

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